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DOCUMENT RESUME ED 380 278 SE 055 878 AUTHOR Cipra, Barry TITLE What's Happening in the Mathematical Sciences, 1993-1994. INSTITUTION American Mathematical Society, Providence, R.I. REPORT NO ISBN-0-8218-8998-2; ISBN-0-8218-8999-0; ISSN-1065-9358 PUB DATE 93 NOTE 109p.; Published annually, starting in 1993. AVAILABLE FROM American Mathematical Society, P.O. Box 5904, Boston, MA 02206-5904 (order no. for volume 1: HAPPENING/lwh, $7; order no. for volume 2: HAPPENING/2WH, $8). PUB TYPE Collected Works Serials (022) JOURNAL CIT What's Happening in the Mathematical Sciences; v1-2 1993-1994 EDRS PRICE MF01/PC05 Pius Postage. DESCRIPTORS Algorithms; Biology; Classification; Coding; Computers; Crystallography; Environmental Education; Geometry; Higher Education; *Innovation; *Mathematical Applications; *Mathematics Instruction; Prime Numbers; Proof (Mathematics); Science Education; Sec.ondary Education IDENTIFIERS *Mathematical Sciences; Medical Technology ABSTRACT This document consists of the first two volumes of a new annual serial devoted to surveying some of the important developments in the mathematical sciences in the previous year or so. Mathematics is constantly growing and changing, reaching out to other areas of science and helping to solve some of the major problems facing society. Volumes and 2 survey some of the important 1 developments in the mathematical sciences over the past year or so. The contents of volume 1 are: (1) "Equations Come to Life in Mathematical Biology"; (2) "Ned Computer Insights from 'Transparent' Proofs"; (3) "You Can't Always Hear the Shape of a Drum"; (4) "Environmentally Sound Mathematics"; (5) "Disproving the Obvious in Higher Dimensions"; (6) "Collaboration Closes in on Closed Geodesics"; "Crystal Clear Computations"; (8) "Camp Geometry"; (7) (9) "Number Theorists Uncover a Slew of Prime Impostors"; and (10) "Map-Coloring Theorists Look at New Worlds." The contents of volume 2 are: (1) "A Truly Remarkable Proof" (Fermat's Last Theorem); (2) "From Knot to Unknot"; (3) "New Wave Mathematics"; (4) "Mathematical Insights for Medical Imaging"; (5) "Parlez-vous Wavelets?" (6) "Random Algorithms Leave Little to Chance"; (7) "Soap Solution"; (8) "Straightening Out Nonlinear Codes"; (9) "Quite Easily Done"; and (10) "(Vector) Field of Dreams." (MKR) ****************************************,---******************** Reproductions supplied by EDRS are the best that can be made from the original document. ******************************************************************** I a "PERMISSION TO REPR /DUCE THIS MATERIAL HAS BEEN 3P/ NTED BY F e -rit TO THE EDUCATIONAL RESOURCES INFORMATION CENTER (ERIC) BEST COPY AVAI U S DEPARTMENT Of EDUCATION Ont. of Educational Rsearch and Imororhorerit ( EDUCATIONAL RESOURCES INFonw4TiON CENTER (ERIC) ljr Ns document has been reproduced 5$ received from th, person or Of(KtM111110,1 OncHrilliong It 0 Minor changes have been made to improve ramoduchon (lushly Point; Of wew Or opinions staled In this docu- mint do not heteasanly rorenl official sok°, OERI position or Introduction Welcome to the inaugural issue of What's Happening in the Mathematical Sciences! To be published annually. What's Happening surveys some of the important developments in the mathematical sciences over the past year or so. Mathematics is constantly growing and changing. reaching out to other areas of science and helping to solve some of the major problems facing society. Here you can read about the development of a mathematical model of the human heart, the solution to a longstanding mathematical problem about the way a drum's shape affects its sound, and the contributions mathematics is making to the solution of environmental problems. What's Happening in the Mathematical Sciences aims to inform the general public about the beauty and power of mathematics. The American ivlathematical Society is pleased to present this new publication. We hope you enjoy it! Samuel M. Rankin. III AMS Associate Executive Director and Director of Publications Cover Illustration. A group of scientists at Los Alamos National Laboratory have developed a mathematical model of ocean dynamics for massively parallel computers that they hope will improve understanding of the role of oceans in global climate change. The colors in this computer-generated picture indicate sea surface temperature from cold (blue) to warm (red). Figure courtesy of Richard Smith. Jchr Dukowicz, and Robert Malone at Los Alamos National Laboratory. 3 Contents Equations Come to Life in Mathematical Biology 3 ISBN 0-8218-8999-0. ISSN 1065-9358 Mathematicians are working with biologists to delve into some of the most challenging problems in biology today, from understanding the human immune system to "computing" the human heart. C 1993 by the American Mathemati:al All rights reserved. New Computer Insights from "Transparent" Proofs 7 Permission is granted to make and distribute verbatim copies of this publication or of indi- Can a computer be trusted when it produces a proof so long and complicated that no human can check vidual items from this publication provided the the details? Theorists have cooked up a new way to tell whether or not a computer proof is right. copyright notice and this permission notice are preserved on all copies. You Can't Always Hear the Shape of a Drum 13 Permission is granted to copy and distribute Can you hear the shape of a drum? is a famous problem that asks if two drums that look different can modified versions of this publication or of indi- make the same sound. A fter decades of head-scra tching, mathematicians have come up with the answer. vidual items from this publication under the conditions for verbatim copying. provided that Environmentally Sound Mathematics 17 the entire resulting derived work is distributed Mathematicians have been teaming up with scientists to work on solving environmental problems, from under the terms of a permission notice identical ocean modeling to dealing with hazardous waste. to this one. 1991 Mathonatics Subject Classification: Pri- Disproving the Obvious in Higher Dimensions 21 mary 00A06. Intuition about our three-dimensional world can be surprisingly misleading when it comes to higher Printed in the United States of America. dimensions, as two recent results in geometry show. Collaboration Closes in on Closed Geodesics 27 An unusual blend of differential geometry and dynamical systems has led to an important theoretical This publication has been typeset using the result about the number of closed "geodesic" curves on distorted spheres. TEX typesetting system running oh a Sol- bourne 5/502 Unix computer. Halftones were created front original photographs with Adobe Crystal Clear Computations 31 Photoshop and illustrations were redrawn us- Growing crystals on a computer? Mathematicians are helping materials scientists to better under ing Adobe Illustrator on Macintosh Quadra stand the nature of crystals, while picking up some challenging mathematical problems along the way. and Macintosh I Ici computers. PostScript code was generated using dvipa by Radical Eye Camp Geometry 35 Softy, are. A group of talented and inquisitive undergraduates "camped out" last summer at the Geometry Center. Typeset on an Agfa/Compugraphic 9600 laser Using sophisticated computer graphics and their own imaginations. they came up with some fascinating imagesetter at the American Mathematical So- mathematics. ciety. Printed at E. A. Johnson. East Provi- dence, RI, on recycled paper. Number Theorists Uncover a Slew of Prime Impostors 39 Strange as it may sound, there arc composite numbers that "masquerade" as primes. A group of tot mathematicians trying to hunt down these prime impostors ended up proving there are infinitely many of them. Map-Coloring Theorists Look at New Worlds 43 How many colors are needed to distinguish neighboring countries on a map? The famous Four Color Theorem notwithstanding, this is a challenging problem in graph theory especially when your maps aren't nat. -1 pulmonary veins aortic valve leaflets left (3) atrium mitral valve leaflets (2) left ventricle chordae right tenclineae ventricle the human heart; the computed/low Researchers (It the Courant Institute of Mathematical .S'cience.v have (mated a three-dimensional mod, i pattern of blood is shown alm,. Grayish lines depict heart fibers. and black spots depict blood. The recent motion of the blood is indicated by .vhowing the.flow pattern the dark lines trailing behind the black spots. The figure shown is a single franic in the simulation of the blood Charles Peskin and David McQueen. just after ventricular dectif,n. Figure courtesy WHAT'S HAPPENING IN THE 2 NIATIIENIATICAI, SCIENCES Equations Come to Life in Mathematical Biology he Nile crocodile and the Egyptian plover have a fascinating relationship. Biology has a host of The croc, ordinarily a surly saurian, will sit placidly on the muddy river problems that call out for bank. mouth wide open, while the bird hops from tooth to tooth scarfing leeches mathematical analysis, and other tasty morsels. Crocody lus niloticus enjoys a thorough oral prophylaxis: from the folding of pro- Pluvianus cum fits gets a meal. teins inside an individual The technical term is .symhiosis. cell to the complex food Something like that is evolving between biologists and mathematicians. Biology webs on the ocean floor. has a host of problems that call out for mathematical analysis. from the folding of proteins inside an individual cell to the complex food webs on the ocean floor. Mathematics. for its part. provides a quantitative framework that can bring order to the organic chaos of nature and point toward new directions for research. Mathematics has brought new insights into biology: biology has inspired new mathematical results. Which you regard as the bird and which the crocodile is a matter, shall we say, of taste. Mathematics and biology are not exactly newcomers to each other. Mathemat ical methods have long been used in population studies, epidemiology, genetics, and physiology, to name a few. And biological problems have spurred the creation of many mathematical techniques, including, arguably. the entire field of statistics. What's new is the depth of detail that mathematical models arc now striving and the attendant depth of theory required. The problems being tackled for today call for closer cooperation than ever before between mathematics and biology. Increasingly. mathematicians are getting in on the ground floor of biological research. working directly with biologists to help tease out the mathematical structure in phenomena ranging from the undulating motion of fish to the beating of the human heart. "The field's very different now than it was thirty years ago,- says Alan Perelson. a mathematical biologist at Los Alamos National Laboratory and president of the Society for Mathematical Biology. "Early mathematical biology was really mathematics with a little inspiration from biology." There ';as little real communication between the fields. But the current generation of mathematical biologists, Perel- son says, consists of researchers "who've been driven by the biology, who look at the details, talk to experimentalists, and generate models that are attempting to answer questions of interest to experimentalists.- Perelson's own work has been in theoretical immunology. He and others in the field are trying to develop mathematical models for the sequence of events that begins when. say, you step on a rusty nail, from the first antigenic signals pre- sented by the invading bacteria, to the final chemotactic processes that close the wound either cleanly or with a lasting scar. It's not just a matter of programming a computer to do a bunch of calculations. Researchers first have to identify the crucial biological aspects of the process and then find the appropriate mathmat- ical equations that describe them. Developing such a thorough understanding. WHAT'S HAPPENING IN THE MATHEMATICAL SCIENCES 6 Perelson says. is the "grand goal" of theoretical immunology. but that goal is still a long way off "We are really at the very beginning. J One reason for that is the daunting complexity of the immune system. The body's response to the variety of pathogens it encounters is, among other things. a pattern-recognition problem: The body must somehow identify an invading virus or bacteria based oa the invader's distinctive pattern of chemical clues. The irn- mune system's ability to do this, researchers believe. depends on the diversity of its ( receptors. "To do pattern recognition [for the immune system] seems to require on the order of ten million different types of receptors,- Perclson explains, "So to to recognize pathogens understand in a profound sense how the system operates one really has to deal with systems of enormous complexiiy.- Math- and respond ematics enters the picture as a tool for modeling not only the individual receptors, but also the overarching structure that enables them to act in concert. The emergence of organized behavior from a collection of individual entities is not unique to the immune system: it is a hallmark of living systems. A central problem in biology is to deduce how properties of a system at one level of orga- nization produce behavior at higher levels -for example, how does the electrical activity in the nervous system of a centipede organize itself into the correct patterns to make the critter's legs move in a coordinated fashion? (Photo by Janet Cole- Nancy Nancy Kopell. a mathematician at Boston University, likens this problem to man.) the task of figuring out how a television works knowing only the properties of transistors. She sees the modeling of "emergent behavior" as a central concern for mathematical biology. "There are many questions in biology involving the behavior of systems in which what you can measure easily... is the behavior of some of the components of the system.- Kopell says. "What you can't easily, or sometimes not at all, get from direct measurements is what's going to happen when you hook all these things up. For that you really need some kind of theory.- Kopell and her mathematical colleague Bard Ermentrout of' the University of Pittsburgh have collaborated with biologists to study the rhythmic neuronal patterns that give rise to swimming in an eel-like fish called a lamprey. Researchers had known for some time that the electrical activity in the lamprey spinal cord something like a could be represented mathematically as a "chain of oscillators- set of pendulums hooked together by springs, but with quite different mathematical properties. Kopell and Ermentrout formulated a new mathematical model based Their model on a deeper analysis of how the oscillators arc hooked together. produced predictions which could be verified by experimentalists, and it provided new insight into how the electrical activity organizes itself to produce the swimming motion in lampreys. The model also helped point nut new directions for biological research. And as new data from rim experiments is found. Kopell and Ermentrout continue to refine their mathematics to better reflect the biology. Computer simulation figures prominently in many of the modeling efforts in mathematical biology. Indeed, revolutions in both hardware and software have been crucial to advances across the board. The Human Genome Project. with its ambitious goal of mapping the roughly three billion base pairs that constitute human DNA, would be inconceivable without machines and mathematical algorithms for dealing with vast amounts of data. ( It's not just a question of storing three billion pieces of information: it's a question ofanalr:ing that data.) Likewise, mathematics is at the heart of much of medical imaging. including CAT scans, nuclear magnetic resonance. and positron emission tomography. These techniques .1.111! WHAT'S HAPPENING IN THE 4 NIATHENIATICAL SCIENCES 7 are made possible by machines that carry out mathematical manipulations of the Mathematics is at the data that pour into them. heart of much of medical One notable example of the use of mathematics and computer simulation in physiology is the work of Charles Peskin and colleagues at the Courant imaging, including CAT Institute of Mathematical Sciences at New York University. They are in the process of scans, nuclear magnetic building a realistic three-dimensional mathematical model of the human heart. resonance, and positron The model. they hope. will give researchers insight into the functioning and emission tomography. malfunctioning of real hearts and lead to improved designs for artificial valves and other replacement parts. "It's a very large effort. and it's still going on," Peskin notes. The model is nearly complete anatomically. but "we're still working on getting the physiology right.- he adds. That means figuring out the appropriate elasticities of the parts. how fast they should contract, and how fast they should reit- x. and then fine-tuning the equations to reflect these physiological attributes. The geometry of the heart is also a crucial part of the model. Conceptually. the Courant heart consists of hundreds of closed curves representing muscle fibers. "In effect the [model] heart is constructed out of a very large array of rubber bands,- Peskin explains. Mathematically, the curves are represented by a string of discrete points, with specified spring-like elasticity between each pair of consecutive points. A computer keeps track of all these points on the order of '3. million of them and immerses them in a computer-simulated bath of blood. Then the real calculation begins: The numerical heart starts to beat. The mathematics of the calculation can be described by something that sounds like the title of a 1950s Japanese monster movie: Hookes's Law Meets the Navier- Stokes Equation. Hookes's Law is the force-displacement relation for springs. familiar from high-school physics: a fancier, nonlinear version of it is used to Detail from the three-dimensional Courant heart. showing the ilwee leaflets of the 'node/ aortic. wive in its closed position. Thefiber architecture of the valve has a fractal structure which has heen predicted here hr solving an equation fOr the mechanical equilibrium of the.fibers under a pressure load. (Illustration created at the Pittsburgh .S.upercomputing Center WHAT'S HAPPENING IN THE MATHEMATICAL, SCIENCES equation, while less familiar, model the heart's muscle fibers. The Navier-Stokes kind, from blood is even more universal: It describes fluid flow of virtually any With the concurrent rev- of the earth's atmosphere. pumping through the heart to global circulation patterns olutions in both biology determine the complex mo- These are the basic mathematical ingredients that and applied and compu- Unfortunately, you can't it. tions of the heart and the blood moving through tational mathematics, IPe- the solutions precisely- sit down with pencil and paper and solve the equations skini says, "the kinds of approximation. And that turns out to be a are only approachable by computer problems that we can re- flow problems is always formidable task, even fbr a supercomputer. Solving fluid particular challenge: tically hope to do are computationally demanding, but the heart model presents a the boundary of the Unlike flow down a pipe or past a spinning turbine, where expanding tremendously." motion of the heart wall is fluid is fixed or moving in a prescribed manner. the solved for. among the unknowns that must be don't know where the "You not only don't know the boundary velocity. you who has collaborated boundary is," notes David McQueen. a mechanical engineer traditional engineering approach is with Peskin for the past fifteen years. "Your going to be hard pressed to solve this problem." for what he calls "im- Instead. Peskin has developed mathematical techniques is that it allows you mersed boundary'' problems. "The beauty of this method don't know toe boundary motions in to do computing in situations where you only application. "The advance.- says McQueen. Modeling heartbeats is not the "and it has al- technique is generally useful in biofluid dynamics." Peskin says. platelet aggregation ready been applied to a wide variety of problems such as propagation in the during blood clotting. aquatic animal locomotion, and wave study of flow in collapsi- inner ear." Peskin anticipates future applications in the (kidney) tubules. and the ble tubes such as thin- walled blood vessels. flow in renal possibilities. such as the flight of birds and bats. There are even nonbiological efficient sails and parachutes. design of aerocR that Peskin The current heart model is a step up from a two-dimensional heart the 2-D heart is began developing in the early 1970s. Paradoxically. Peskin notes. That's mainly because the still, in some ways, more realistic than the 3-D model. for now has forced the modelers extra effort of computing in three dimensions advances in both theory and hardware to use a simpler muscle model. Further David McQueen and Charles Peskin. 2-D heart is likely to will undoubtedly bring the 3-D model up to speed, but the Photo reffinted /11th permission of really like is to continue being used for experimental computations. "What we'd Projects in Scientific Computing. Pittv- rough results, and then perhaps do a few burgh Superromputing Center use the 2-D model as a way of getting 3-D computations to verify those findings,- McQueen says. valve design. Indeed, the 2-1) model has already proved useful in artificial heart h the shape of a prosthetic mitral valve (the gate between the By experimenting v. that simultaneously left atrium and ventricle). McQueen and Peskin found a design drop across increased the flow velocity near the valve and reduced the pressure et in clinical use, While not two features that are prized in artificial valves. it the design has been patented and licensed. likely to The 3-D model has not yet had any such applications, but those are physiologically realistic and as the computing come as the model becomes more upwards of fifty demands get more manageable. (Currently a single beat takes other models in hours of supercomputer time.) Peskin sees the heart model, and revolutions in the future, as important experimental tools. With the concurrent "the kinds of both biology and applied and computational mathematics, he says. problems that we can realistically hope to do are expanding tremendously." WHAT'S HAPPENING IN THE 6 NIXTHENIATICAL SCIENCES New Computer Insights from "Transparent" Proofs athematicians are professional skeptics. When told of a new result. Today's lightning-ast, their first response is, Where's the proof? Even when shown a proof, high-tech adding machines they're not completely convinced it's correct until they check every last line. take the labor out of long, This professional skepticism isn't limited to traditional mathematical proofs. It laborious calculations. But extends to results produced by ,:omputers as well. Today's lightningfast, high- they leave behind the lin- tech adding machines take the labor out of long. laborious calculations. making it possible to carry out computations that could never be done by hand. But they gering question, Did the leave behind the lingering question. Did the computer do its job correctly? computer do its job cor- A sequence of recent breakthroughs in theoretical computer science may put rectly? that question to rest. Researchers have found some unexpected new ways by which computers can prove "beyond a shadow or a doubt" that the results they provide are indeed reliable. Moreover, these developments are giving theorists new insights into some of the hardest problems of computer science. Guaranteeing the reliability of computer results is obviously of concern to more than mathematicians. But by thinking of computations themselves as proofs that certain inputs produce certain outputs, theoretical computer scientists are able to view anything a computer does in logical mathematical terms. Moreover, the computational aspects of many problems can be recast as purely mathematical questions in areas such as graph theory or elementary. first-order logic. The abstract language of mathematics helps daffy the essential issues, which might otherwise be lost among the details of individual applications. Some computations are easy enough to check. For example. researchers often need to know if them- is a path that travels along the edges of a graph, visit;ng each vertex once and only once -what graph theorists call a "Hamiltonian cycle." (This kind of problem crops up in applications such as designing efficient telecom- Figure la. The dad% edges "prove" that munications networks.) If a computer says there is a Hamiltonian cycle, it can this graph has a Hamiltonian cycle prove it simply by pointing out the path (as done with dark lines in Figure I a). But when it says there is no such path for the graph in Figure 1b, how can you be sure it didn't overlook one -or, worse, that your computer saw one but chose not to tell you? The computer can. of course, produce a proof by trying all possible routes around the graph and showing that none is a Hamiltonian cycle. That's not an unreasonable thing to do for Figure lb. But the number of possible routes grows so quickly with the number of vertices that this straightforward approach soon becomes unwieldy. For graphs that typically occur in telecommunications network problems. for example, this kind of proof would take inconceivably long even on the fastest conceivable supercomputer. That defeats the purpose of having a fast machine. Worse, one is still left with the task of checking that all the computations were done correctly. The problem is. errors in a proof don't a]ways, or even usually. call attention to themselves----and all it takes to invalidate an entire proof' is one mistake, as minor Figure lb. /his figure does not hare a as a misplaced minus sign. "Mathematical proofs are very fragile," says Laszlo or does it? Hamiltonian cycle Babai, a theoretical computer scientist at the University of Chicago. Like a string 1111 IMIIMMV WHAT'S HAPPENING IN nip: 7 NIATHENIATICAL SCIENCES 10